Turk J Phys
(2014) 38: 253 – 260
Turkish Journal of Physics
http://journals.tubitak.gov.tr/physics/
c TÜBİTAK
⃝
doi:10.3906/fiz-1402-4
Research Article
Neutron-rich
208
Pb nucleus with delta excitation under compression
Mohammed Hassen Eid ABU-SEI’LEEK∗
Department of Physics, Faculty of Sciences, Majmaah University, Zulfi, Saudi Arabia
Received: 10.02.2014
•
Accepted: 14.05.2014
•
Published Online: 11.06.2014
•
Printed: 10.07.2014
Abstract:The spherical Hartree–Fock approximation is applied to a no-core shell model with a realistic effective baryon–
baryon interaction. The ground state properties of a heavy spherical neutron-rich doubly magic
208
Pb nucleus under
compression are investigated. It is found that the nucleus becomes more bound with the occurrence of ∆ resonances.
The creation of ∆ increases as the compression is continuous. There is a considerable reduction in the compressibility
when the ∆ degree of freedom is activated. It is found that the ∆ particle is the basic component of the
208
Pb nucleus
besides nucleons at the ground state without any compression. When the nucleus is compressed to about 4.31 times
the ordinary density, the ∆ component is sharply increased to about 14.4% of all baryons in the system. It is found
that there is a radial density distribution for ∆ at the ground state of
208
Pb nucleus without any compression. The
single particle energy levels are calculated and their behaviors are examined under compression too. A good agreement
was obtained between the results of the effective Hamiltonian and the phenomenological shell model for the low lying
single-particle spectra.
Key words: Nuclear structure, compressed finite nuclei, ∆ -resonance, Hartree–Fock method, shell model, heavy
spherical doubly magic
208
Pb nucleus
PACS: 21.10.Dr; 21.60.Cs; 21.60.Jz; 24.30.Gd; 27.80.+w
1. Introduction
Doubly magic nuclei have been the cornerstone of nuclear structure since the inception of the nuclear shell
model [1]. The properties of the spherical doubly magic
208
Pb nucleus are the focus of nuclear structure studies
[2].
The structure of nuclei at intermediate and high energies is importantly understood by the excitation of
∆ isobars. Various probes are formed to provide an exciting challenge both theoretically and experimentally,
especially in the search for constructive and coherent pion production [3,4]. Current interest in understanding
the collision of both light and heavy ions [5–10] involves ∆ excitation and its decay to nucleon and to pions.
The heavy ion collision problem [11] and astrophysical phenomena such as supernova explosions or structure
of neutron stars [12] are understood by investigating the delta formation in the nucleus, as a function of
compression. The ground state properties of the spherical heaviest doubly magic 208 Pb nucleus and the
formation of the ∆ isobars at equilibrium under large radial compression are studied by the present work.
The effect of ∆ resonance on the ground state properties of closed shell nuclei is emphasized. By using a
spherical mean field method, the calculations are performed. In other words, the constrained spherical Hartree–
∗ Correspondence:
[email protected]
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Fock (CSHF) approximations with a model space of the 8 space (9 major oscillator shells) and realistic effective
Hamiltonian [13] with N–N, (N–∆), and (∆–∆) interactions are used. By using the Brueckner G-matrix
[14–19] and its adopted improvement [20–22], the effective baryon–baryon interaction is evaluated.
The results of the role of ∆’s in finite nuclei have been investigated in a 6-model space (7 shells) [23–
35]. A collection of nucleons and ∆-resonances is considered to form the nucleus. The effects of including
the ∆-degrees of freedom on the Hartree–Fock energy, density distribution, and ∆ -orbital occupations in the
ground state and under large amplitude static compression at temperature T = 0 have also been examined.
The selected nuclei were
16
O,
40
Ca,
56
Ni,
90
Zr,
100
Sn, and
132
Sn.
In the present work, the main goal extends these results to a new region of the periodic table. The previous
effects with different adjusting parameters are considered in a large model space for the heavy doubly magic
208
Pb nucleus. The emphasis is on single particle energy levels for nucleons and deltas with large amplitude
static compression in a model space consisting of 9 major oscillator shells with CSHF approximation. The
calculation was done with the use of a realistic effective Hamiltonian with different potentials. The Brueckner
G-matrices that are used are generated from coupled channels NN, N ∆ , and π NN. This is done to give a
good description of NN data up to 1 GeV. By using the same method reported before [36,37], the effective
interactions of the nuclear shell model are calculated. It is a good tool to study highly compressed nuclei at
densities accessible to relativistic heavy ion collisions.
The mean-field calculations demonstrated the effective Hamiltonian, H ef f , and the calculation procedures. Based on the study presented in the literature [38–40], the 2-body matrix elements are scaled in the
N–N sector to an optimal value of ℏω ′ , the oscillator energy for 208 Pb nucleus in the 9 major oscillator shells
with the 10 delta orbits. By using adjusting parameters (λ1 , λ2 , and ℏω ′ ) , it is possible to obtain such a fit to
the equilibrium binding energy and rrms radius. In this work, the adjusting parameters (λ1 ,λ2 , and ℏω ′ ) are
shown in the Table. This paper is organized as follows: section 2 contains the results and discussion and the
conclusion is presented in section 3.
Table. Adjusting parameters of effective Hamiltonian for
208
Pb for the model space of 9 oscillator shells for which the
calculations were performed. The binding energy (point mass rrms ) that was fitted was –1636.5 MeV (5.503 fm) for
208
Pb 44].
Nucleus 208 Pb
(9 shell)
λ1
0.997
λ2
1.001
ℏω ′ (MeV)
7.345
2. Results and discussion
The total number of ∆’s is zero in the ground state at equilibrium (zero constraint) in the case of the equal
doubly magic
16
O,
40
Ca,
56
Ni,
90
Zr, and
state for the neutron-rich doubly magic
100
132
Sn nuclei. Moreover, the number of ∆ ’s is not zero in the ground
Sn nucleus. These results give us strong motivation to study the
208
heavy neutron-rich doubly magic
Pb nucleus in a larger model space consisting of 9 major oscillator shells
for nucleons and 10 orbits for ∆ ’s, making a total of 47 baryons orbits.
Figure 1 displays the Hartree–Fock energies EHF versus rrms using RSC potential for 208 Pb. In this
figure, there is virtually no difference in the results with and without ∆ ’s at equilibrium. It is seen that without
the ∆ -degree of freedom in the system, EHF increases steeply towards zero binding energy under compression.
When the transition to ∆ is allowed (dashed curve), the nucleus remains bound as density is increased to 4.31
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ABU-SEI’LEEK/Turk J Phys
of normal density. This shows about 1005.6 MeV and 627.3 MeV of excitation energy to achieve a 49% volume
reduction in the nucleon-only results, and nucleons and ∆ ’s results, respectively.
It appears from the above results that 627.3 MeV of excitation energy is enough to reduce the volume by
49% and the energy by 38% more. This suggests that the less dense outer part of the nucleus initially responds
to the external load more readily than the inner part. The results show that there is a significant reduction in
the static compression modulus for RSC static compressions, which is reduced by including the ∆ excitations.
The consequence of this reduction is the softening of the nuclear equation of state at larger compression. To see
the role of ∆ in determining the equation of state, Figure 1 shows the dependence of EHF on the compression
characterized by rrms . It can be seen that near equilibrium (rrms = 5.50 fm) all curves agree. Furthermore,
the inclusion of ∆ orbits tends to decrease EHF for compressed nuclei. The role of the ∆ ’s is less significant
as rrms approaches the ground state value.
Figure 1 shows that as the static load force increases the compression of the nucleus with nucleons only
is less than that of the other nucleus with nucleons and ∆ ’s.
In terms of relativistic heavy-ion collision, Figure 1 implies that the heavy nucleus can more easily
penetrate when the ∆ degree of freedom becomes explicit.
Figure 2 shows that the number of ∆’s increases rapidly as volume decreases. It is interesting to note
in Figure 2 that the numbers of ∆0 ’s and ∆+ ’s are the same until rrms = 3.90 fm. Beyond this radius, the
increment in the number of ∆0 ’s is larger than the number of ∆+ ’s as the compression is continuous. This is
due to the fact that the number of neutrons is greater than the number of protons in the
–400
–600
N–only
N+ Δ's
Number of Δ's
EHF (MeV)
–800
208 Pb
(9 shells)
–1000
–1200
–1400
–1600
–1800
3.0
3.5
4.0
4.5
rrms (fm)
5.0
5.5
Figure 1. CSHF energy as a function of the point mass
rrms for 208 Pb evaluated in 9 major oscillator shells with
10 ∆ -orbitals. The dashed curve corresponds to CSHF
calculations including the ∆ ’s, while the solid curve corresponds to CSHF with nucleons only.
32
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
208
Pb nucleus.
ΔT =Δ++ Δ0
208
Pb
(9 shells)
ρ=4.3 ρequilibrium
Δ0
Δ+
3.0
3.5
4.0
4.5
rrms (fm)
5.0
5.5
Figure 2. Number of ∆ ’s as a function of rrms for 208 Pb
in 9 major shells model space. The upper curve is for the
total number of ∆ ’s. The dotted curve is for the number
of ∆+ and the dashed curve is for ∆0 .
Although there is a rapid rise in the ∆-population as compression increases, the change in the total
number of ∆ ’s is 0.144. It is interesting to note that there is a consistency of the amount of N– ∆ mixing
with the amount of excitation energy exhibited with compression. That is, when 0.144 ∆ ’s are presented,
the excitation energy is in the order of 0.144(M-m) ≈ 42.77 MeV. Thus, on the scale of the unperturbed
single-particle energies, a substantial fraction of the compressive energy is delivered through the N ∆ and ∆ – ∆
interactions to create more massive baryons in the lowest energy configuration of the nucleus. In other words,
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ABU-SEI’LEEK/Turk J Phys
the number of ∆’s can be increased to about 30 at rrms = 3.38 fm, which corresponds to about 4.31 times the
normal density.
Figure 2 shows that the number of created ∆’s increases sharply when the 208 Pb nucleus is compressed to
a volume of about 0.77 of its equilibrium size. However, at this nuclear density, which is 4.31 times the normal
density, the percentage of nucleons converted to ∆ is only about 14.4% in 208 Pb. This result is consistent
with the information extracted from the data of relativistic heavy-ions collisions. In some heavy-ion collision
experiments, the ∆ ’s may constitute up to 10% of the nuclear constituents when the system is compressed
[41,42].
The calculations of Figure 2 show that the number of ∆ ’s is not zero at equilibrium (zero constraint) in
the
208
Pb nucleus. This result is consistent with the previous finding for
132
Sn.
Therefore, the heavy nuclei are a composite of the ∆’s besides the nucleons at equilibrium ground state
without any compression.
Because of the limitations of the model space, our calculations for higher densities are more speculative.
Nevertheless, they can give us some idea about how the ∆ population can be increased as the nucleus is
compressed to higher densities accessible to relativistic heavy-ion collisions.
The radial density distribution for
Figure 3.
208
Pb at equilibrium state without any compression is displayed in
0.20
208Pb
(9 shells)
rrms =5.503fm
Equilibrium Case
ρT
ρ (baryons/fm3 )
0.15
ρn
0.10
ρp
0.05
0.00
ρΔ (Scaled by 200)
0
1
2
3
4
5
6
r (fm)
7
8
9
10
Figure 3. Total ρT , proton ρp (dashed line), neutron ρn (dotted line), and delta ρ∆ (solid line) radial density distribution
for
208
Pb at equilibrium state without any compression in a model space of 9 major oscillator shells.
It can be seen from this figure that there is a radial density distribution for ∆ at the ground state of the
208
Pb nucleus without any compression. This emphasizes the important fact that the ∆ particle is the basic
component of heavy nuclei besides nucleons (protons and neutrons) at ground state without any constraints.
Our results shown in this figure are consistent with the results of figure 2 in Zuo et al. [43] that shows many
methods to find the radial density of
208
Pb.
Figure 4 shows the radial density distribution under compression. The neutron radial density is higher
than the proton density at all values of r. This is due to Coulomb repulsion between the protons. The ∆ -radial
density distribution, under high compression (point mass rrms = 5.05 fm), reaches a peak value of about 0.02
of the proton (or neutron) radial density at r = 3.1 fm. ∆ -mixing with the nucleons in the 0p 3/2 , 0p 1/2 ,
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ABU-SEI’LEEK/Turk J Phys
0d 5/2 , 0d 3/2 , 0f
7/2 ,
0f
5/2 ,
1p 3/2 , 1p 1/2 , 0g 9/2 , and 0g 7/2 orbitals occurs and this explains the shape of the
∆-radial distribution presented in Figure 4.
Figure 5 displays the radial density distributions of 208 Pb, which is evaluated of about 0.23 reduced
volumes. In this case, the ∆ -radial density distribution reaches a peak value of about 0.95 of the proton radial
density.
0.30
0.20
ρT
ρT
0.18
0.25
Pb
(9 shells)
rrms =5.047fm
0.14
0.12
ρn
0.10
0.08
ρp
0.06
0.04
Pb
(9 shells)
rrms =3.381fm
0.20
0.15
ρn
0.10
ρp
0.05
0.02
0.00
208
208
ρ(baryons/fm3 )
ρ(baryons/fm3)
0.16
ρΔ (Scaled by 10)
0
Figure 4.
1
2
3
4
5
6
r (fm)
7
8
9
0.00
10
0
Total ρT , proton ρp (dashed line), neutron
ρn (dotted line), and delta ρ∆ (solid line) radial density
208
ρΔ
distribution for
Pb at point mass rrms = 5.047 fm in a
model space of 9 major oscillator shells.
Figure 5.
1
2
3
4
5
6
r (fm)
7
8
9
10
Total ρT , proton ρp (dashed line), neutron
ρn (dotted line), and delta ρ∆ (solid line) radial density
distribution for 208 Pb at point mass rrms = 3.381 fm in a
model space of 9 major oscillator shells.
It can be seen from Figures 4 and 5 that as compression increases the total radial density increases and
the radial density distribution of ∆ ’s increases sharply, but the radial density of nucleons decreases sharply.
Figure 6 shows the total radial density for
208
Pb in 9 oscillator shells at large compression (rrms = 3.38
fm) and at equilibrium (point mass radius rrms = 5.50 fm). This figure shows that when the volume of the
nucleus is decreased by 0.77 of the equilibrium volume the radial density is increased by 1.65 of its value at the
equilibrium case.
Clearly, the density in the interior rises relative to the interior density at equilibrium as the nucleus is
compressed. This is in contrast to the behavior of the radial density on the outer surface, where the radial
density distribution is higher at equilibrium than the radial density when the static load is applied.
In Figure 7, the lowest neutron single particle energy levels as a function of rrms are displayed. The
orbits curved up as the load on the nucleus increased. This is because the kinetic energy of the baryons, which
is a positive quantity, becomes more influential than the attractive mean field of the baryons.
The single particle energies increase linearly after rrms = 4.0 fm; they intersect at higher compression
especially for higher orbitals, and consequently the nucleus becomes unbound.
The behavior of single particle energy levels has good agreement with the orbital ordering of the standard
shell model. The gap is very clear between the shells. The splitting of the levels in each shell is an indication
that L–S coupling is strong enough in RSC potential, i.e. L–S coupling becomes stronger as the static load on
the nucleus increases. When the nucleus is compressed, the splitting of the orbitals becomes clearer, especially
in delta orbitals.
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ABU-SEI’LEEK/Turk J Phys
0.30
ρ (baryons/fm3 )
0.25
Single particle energies(MeV)
rrms=3.381fm
208
Pb
(9 shells)
0.20
0.15
rrms=5.503fm
Equilibrium
0.10
0.05
0.00
0
1
2
3
4
5
6
r (fm)
7
160
140
120
100
80
60
40
20
0
–20
–40
–60
–80
–100
–120
8
9
10
Figure 6. Total radial density distribution for 208 Pb at
equilibrium ( rrms = 5.503 fm) (dashed line) and at rrms
208
Pb
(9 shells)
3.0
3.5
Figure 7.
states for
208
4.0
4.5
rrms (fm)
5.0
0s 1/2
0p 3/2
0p 1/2
0d 5/2
1s 1/2
0d 3/2
0f 7/2
1p 3/2
0f 5/2
1p 1/2
5.5
Single particle energy of lowest 10 neutron
Pb in 9 oscillator shells as a function of rrms .
= 3.381 fm (solid line).
Further efforts will concentrate on extending this study to a large model space for heavier nuclei and
alternative potentials in order to investigate the sensitivity of nuclear properties to the N– ∆ and ∆ – ∆
interactions. One ultimate goal is to predict the behavior of nuclear systems at finite temperature with the
inclusion of the ∆ -degree of freedom.
3. Conclusion
The ground state properties of the heavy neutron-rich spherical 208 Pb nucleus have been investigated by using
a realistic effective baryon–baryon Hamiltonian in the constrained Hartree–Fock approximation. It is found
that the nucleus becomes more bound with the occurrence of ∆ -resonances. The nuclear shell model is derived
in this approach with single particle levels occupied by baryons that are a mixture of nucleons and deltas. As
shown in
208
Pb, the results compare favorably with those of the phenomenological successful shell model.
It is found that the ∆ particle is the basis component of the heavy nuclei besides nucleons at ground state
without any compression. There is a considerable reduction in compressibility when the ∆ degree of freedom
is active. The creation of ∆ ’s increases as the compression continues.
Finally, a large fraction of the excitation energy is required to compress the nucleus used to create mass
in the form of ∆ ’s.
Acknowledgment
I acknowledge support for this work from the Engineering and Applied Sciences, Research Center, Scientific
Research Deanship, Majmaah University, under Award No. 10-2012.
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